Integrand size = 41, antiderivative size = 96 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=(g+2 h+5 i) x+\frac {1}{2} (h+2 i) x^2+\frac {i x^3}{3}-\frac {1}{2} (d+e+f+g+h+i) \log (1-x)+\frac {1}{3} (d+2 e+4 f+8 g+16 h+32 i) \log (2-x)+\frac {1}{6} (d-e+f-g+h-i) \log (1+x) \]
(g+2*h+5*i)*x+1/2*(h+2*i)*x^2+1/3*i*x^3-1/2*(d+e+f+g+h+i)*ln(1-x)+1/3*(d+2 *e+4*f+8*g+16*h+32*i)*ln(2-x)+1/6*(d-e+f-g+h-i)*ln(1+x)
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.95 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{6} \left (6 (g+2 h+5 i) x+3 (h+2 i) x^2+2 i x^3-3 (d+e+f+g+h+i) \log (1-x)+2 (d+2 e+4 (f+2 g+4 h+8 i)) \log (2-x)+(d-e+f-g+h-i) \log (1+x)\right ) \]
(6*(g + 2*h + 5*i)*x + 3*(h + 2*i)*x^2 + 2*i*x^3 - 3*(d + e + f + g + h + i)*Log[1 - x] + 2*(d + 2*e + 4*(f + 2*g + 4*h + 8*i))*Log[2 - x] + (d - e + f - g + h - i)*Log[1 + x])/6
Time = 0.35 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {2019, 2462, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(x+2) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{x^4-5 x^2+4} \, dx\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{x^3-2 x^2-x+2}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {d+2 e+4 f+8 g+16 h+32 i}{3 (x-2)}+\frac {-d-e-f-g-h-i}{2 (x-1)}+\frac {d-e+f-g+h-i}{6 (x+1)}+g \left (\frac {2 h+5 i}{g}+1\right )+x (h+2 i)+i x^2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \log (1-x) (d+e+f+g+h+i)+\frac {1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac {1}{6} \log (x+1) (d-e+f-g+h-i)+x (g+2 h+5 i)+\frac {1}{2} x^2 (h+2 i)+\frac {i x^3}{3}\) |
(g + 2*h + 5*i)*x + ((h + 2*i)*x^2)/2 + (i*x^3)/3 - ((d + e + f + g + h + i)*Log[1 - x])/2 + ((d + 2*e + 4*f + 8*g + 16*h + 32*i)*Log[2 - x])/3 + (( d - e + f - g + h - i)*Log[1 + x])/6
3.1.84.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\left (\frac {h}{2}+i \right ) x^{2}+\left (g +2 h +5 i \right ) x +\frac {i \,x^{3}}{3}+\left (-\frac {d}{2}-\frac {e}{2}-\frac {f}{2}-\frac {g}{2}-\frac {h}{2}-\frac {i}{2}\right ) \ln \left (x -1\right )+\left (\frac {d}{3}+\frac {2 e}{3}+\frac {4 f}{3}+\frac {8 g}{3}+\frac {16 h}{3}+\frac {32 i}{3}\right ) \ln \left (x -2\right )+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right ) \ln \left (x +1\right )\) | \(99\) |
default | \(\frac {i \,x^{3}}{3}+\frac {h \,x^{2}}{2}+i \,x^{2}+g x +2 h x +5 i x +\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right ) \ln \left (x +1\right )+\left (-\frac {d}{2}-\frac {e}{2}-\frac {f}{2}-\frac {g}{2}-\frac {h}{2}-\frac {i}{2}\right ) \ln \left (x -1\right )+\left (\frac {d}{3}+\frac {2 e}{3}+\frac {4 f}{3}+\frac {8 g}{3}+\frac {16 h}{3}+\frac {32 i}{3}\right ) \ln \left (x -2\right )\) | \(102\) |
parallelrisch | \(g x +i \,x^{2}+\frac {\ln \left (x -2\right ) d}{3}+\frac {2 \ln \left (x -2\right ) e}{3}-\frac {\ln \left (x -1\right ) d}{2}-\frac {\ln \left (x -1\right ) e}{2}+\frac {h \,x^{2}}{2}+\frac {i \,x^{3}}{3}-\frac {\ln \left (x +1\right ) i}{6}+\frac {\ln \left (x +1\right ) f}{6}+2 h x +\frac {\ln \left (x +1\right ) d}{6}-\frac {\ln \left (x +1\right ) e}{6}-\frac {\ln \left (x +1\right ) g}{6}+\frac {32 \ln \left (x -2\right ) i}{3}-\frac {\ln \left (x -1\right ) i}{2}+\frac {8 \ln \left (x -2\right ) g}{3}-\frac {\ln \left (x -1\right ) g}{2}+\frac {4 \ln \left (x -2\right ) f}{3}-\frac {\ln \left (x -1\right ) f}{2}+\frac {16 \ln \left (x -2\right ) h}{3}-\frac {\ln \left (x -1\right ) h}{2}+5 i x +\frac {\ln \left (x +1\right ) h}{6}\) | \(156\) |
risch | \(\frac {i \,x^{3}}{3}+\frac {h \,x^{2}}{2}+i \,x^{2}+g x +2 h x +5 i x +\frac {\ln \left (x +1\right ) d}{6}-\frac {\ln \left (x +1\right ) e}{6}+\frac {\ln \left (x +1\right ) f}{6}-\frac {\ln \left (x +1\right ) g}{6}+\frac {\ln \left (x +1\right ) h}{6}-\frac {\ln \left (x +1\right ) i}{6}-\frac {\ln \left (1-x \right ) d}{2}-\frac {\ln \left (1-x \right ) e}{2}-\frac {\ln \left (1-x \right ) f}{2}-\frac {\ln \left (1-x \right ) g}{2}-\frac {\ln \left (1-x \right ) h}{2}-\frac {\ln \left (1-x \right ) i}{2}+\frac {\ln \left (2-x \right ) d}{3}+\frac {2 \ln \left (2-x \right ) e}{3}+\frac {4 \ln \left (2-x \right ) f}{3}+\frac {8 \ln \left (2-x \right ) g}{3}+\frac {16 \ln \left (2-x \right ) h}{3}+\frac {32 \ln \left (2-x \right ) i}{3}\) | \(180\) |
(1/2*h+i)*x^2+(g+2*h+5*i)*x+1/3*i*x^3+(-1/2*d-1/2*e-1/2*f-1/2*g-1/2*h-1/2* i)*ln(x-1)+(1/3*d+2/3*e+4/3*f+8/3*g+16/3*h+32/3*i)*ln(x-2)+(1/6*d-1/6*e+1/ 6*f-1/6*g+1/6*h-1/6*i)*ln(x+1)
Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{3} \, i x^{3} + \frac {1}{2} \, {\left (h + 2 \, i\right )} x^{2} + {\left (g + 2 \, h + 5 \, i\right )} x + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \]
1/3*i*x^3 + 1/2*(h + 2*i)*x^2 + (g + 2*h + 5*i)*x + 1/6*(d - e + f - g + h - i)*log(x + 1) - 1/2*(d + e + f + g + h + i)*log(x - 1) + 1/3*(d + 2*e + 4*f + 8*g + 16*h + 32*i)*log(x - 2)
Timed out. \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\text {Timed out} \]
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{3} \, i x^{3} + \frac {1}{2} \, {\left (h + 2 \, i\right )} x^{2} + {\left (g + 2 \, h + 5 \, i\right )} x + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \]
1/3*i*x^3 + 1/2*(h + 2*i)*x^2 + (g + 2*h + 5*i)*x + 1/6*(d - e + f - g + h - i)*log(x + 1) - 1/2*(d + e + f + g + h + i)*log(x - 1) + 1/3*(d + 2*e + 4*f + 8*g + 16*h + 32*i)*log(x - 2)
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{3} \, i x^{3} + \frac {1}{2} \, h x^{2} + i x^{2} + g x + 2 \, h x + 5 \, i x + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, {\left (d + e + f + g + h + i\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left ({\left | x - 2 \right |}\right ) \]
1/3*i*x^3 + 1/2*h*x^2 + i*x^2 + g*x + 2*h*x + 5*i*x + 1/6*(d - e + f - g + h - i)*log(abs(x + 1)) - 1/2*(d + e + f + g + h + i)*log(abs(x - 1)) + 1/ 3*(d + 2*e + 4*f + 8*g + 16*h + 32*i)*log(abs(x - 2))
Time = 7.93 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03 \[ \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=x\,\left (g+2\,h+5\,i\right )+\frac {i\,x^3}{3}-\ln \left (x-1\right )\,\left (\frac {d}{2}+\frac {e}{2}+\frac {f}{2}+\frac {g}{2}+\frac {h}{2}+\frac {i}{2}\right )+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right )+\ln \left (x-2\right )\,\left (\frac {d}{3}+\frac {2\,e}{3}+\frac {4\,f}{3}+\frac {8\,g}{3}+\frac {16\,h}{3}+\frac {32\,i}{3}\right )+x^2\,\left (\frac {h}{2}+i\right ) \]